Hidden Fractals Suggest Answer to Ancient Math Problem

Researchers have found a fractal pattern underlying everyday math. In the process, they’ve discovered a way to calculate partition numbers, a challenge that’s stymied mathematicians for centuries.

Partition numbers track the different ways an integer can be divvied up. The number 3, for example, has three unique partitions: 3, 2 + 1, and 1 + 1 + 1. Partition numbers grow so fast that mathematicians have a hard time predicting them.

“The number 10 has 42 partitions, but with 100 you have 190,569,292 partitions. They get impossibly huge to add up,” said mathematician Ken Ono of Emory University.

Since the 18th century, generations of mathematicians have tried to find a way of predicting large partition numbers. Srinivasa Ramanujan, a self-taught prodigy from a remote Indian village, found a way to approximate partition numbers in 1919. Yet before he could expand on the work, and convert it to a clean equation, he died in 1920 at the age of 32. Mathematicians ever since have puzzled over Ramanujan’s manuscripts, which tie the primes 5, 7 and 11 to partition numbers.

Inspired by Ramanujan’s work and that of the late mathematician A.O.L. Atkin, Emory mathematicians Amanda Folsom and Zachary Kent joined Ono to discover an infinite, fractal-like pattern to the series. It is described in a paper hosted by the American Institute of Mathematics.

“It was like living in a darkened home for years, and then finally someone turned on the lights. When Zach and I realized the structure, we knew we were right,” Ono wrote in an e-mail to Wired.com. “We see the same mathematical structures over and over and over again, similar to how you see repeating elements in the Mandelbrot set as you fly through it. That’s why we say they’re fractal,” he said.

In a separate paper, Ono and Jan Hendrik Bruinier of Germany’s Darmstadt Technical University describe a function, deemed “P,” that can churn out any integer’s partition number.

The combined research doesn’t quite reveal a mathematical representation of the universe’s structure, Ono said, but it does kill partition numbers as a way to encrypt computer data.

“Nobody’s ever going to do that now, since we now know partition numbers aren’t random,” Ono said. “They’re completely predictable and we should no longer pretend they’re mysterious.”

The discoveries should help solve similar problems in number theory, but Ono said he’s most excited about closing an exceptionally “frustrating but romantic” chapter in mathematical history.

Video: Zooming in on the Mandelbrot set, a famous fractal that illustrates repeating patterns in an infinite series./YouTube, gooozz.

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